This blog will connect Primary Gifted Program instructors for professional learning and problem-solving.
I went to Chapter 4: How Much Change Will I Get?Einstein LevelProblem 3Page 35Because my daughter and her family live in Canada and I have always let Jennifer figure out the exchange when I visit Toronto, I chose Einstein’s problem concerning the differences between the two currencies: Canadian and United States. Einstein’s problem is not applicable now, however, because for the first time in decades, the American dollar is of lesser value than the Canadian dollar. However, that is beside the point. Using the information given in the statement problem, I figured to buy a ticket to the movies for $10.00 Canadian; I would need $7.50 American. After perusing multiple money problems in Zaccaro’s tome, I feel it is important to also have coins and paper money available for the children. Many will be able to do the problems in their heads, but manipulatives are always a positive reinforcement to learning. Touch, feel and see…the magical trio!Because many of my students travel world-wide, I feel I should have on hand varied currencies (coins and paper) and be able to reconstruct problems, like #3, to include other countries and the United States. This gives the students yet another perspective to problem solving.
Chapter 1 “What’s the Next Number?” Glad Mariah on page 1 made a chart and figured out that she can get an allowance of $512 each week when she is 14. You Go Girl! I like the way the diagram shows the progression of the computation. Great introduction of a pertinent vocabulary word when the cartoon figure talks about “predicting” as an important part of math on page 2. On p. 3 when the author is describing how to split 25 in half, I would like to have seen a visual for the fraction ½ or an explanation of why a fraction is written the way it is, but maybe (in the author’s opinion) it’s unnecessary. The first problem, Einstein Level page 7, I solved by adding the next odd number in the series, which would be 7 for an answer of 25. On page 281 the reader is reminded that the answer of 25 is the next number in the “perfect squares” sequence. A good example of students having two possibilities for arriving at the same answer. The next three problems involve dividing fractions. I’m assuming that even if students don’t understand the mechanics of how to divide a fraction (invert and multiply) they can still follow the “pattern” established for writing a fraction and divide the denominator by the appropriate number and still get the right answer.The fifth problem on page 8 can be solved either by using a diagram listing hours of the day with the weight for each hour doubled from the previous hour or a formula for plugging in the right numbers. Personally, I’m a diagram girl, which gets me into lots of trouble when the problems involve much larger numbers and more involved questions. I find this book to be engaging, fun to read, and challenging (?!) for my rusty math skills.
NanetteG - You incorporated two outstanding suggestions into your answer:1. The use of monetary manipulatives for those who are tactile learners and for those who like to manipulate the concrete and have the “hands-on” experience before moving onto the abstract idea of what each coin/bill represents 2. The idea of using currencies from various countries, which would open up an avenue for students to pursue and design their own problems and activities based on diverse cultures, not only incorporating the value of each coin/bill, but also facts about the countries, products made and grown in those countries, etc…
I like to cook so I selected the "Oh No, I have to Change the Recipe" problems on page 59. I liked problem # 5 (p.59) where Eric lost his train of thought and dumped in a cup more flour than he should of. I've been gulity of doing the same! Needless to say, Eric needs an additional 1/4 cup of sugar and 1/2 cup of oats. Wonder why the problem didn't mention the liquid (milk?) which would also have to be increased.The kids would enjoy any of these if they also involved some cooking - and of course eating!
I completed the Einstein level problem in Chapter 4 “How much change will I get”. I chose #1 on page 35. I worked through the problem using the steps introduced on pages 27-30. And I even got the right answer! I think this kind of problem is sort of intimidating to kids. But after reading this chapter, it doesn’t seem that way. Another problem that I found especially worthwhile was #5 on page 36. The whole change issue is very confusing, but so important to be able to figure out! I would love to have a “store” when doing this chapter and have the kids buy and sell things! Hhhmmm…..
In response to NanetteG – I am a diagram girl, too! I really do better when I can see what is going on. But, like you, I get into some trouble when things get too big. I’ve found this book fantastic at giving other ways to solve problems that are just as concrete for me as drawing a diagram. I don’t hate math so much! Woohoo!
In response to Of Life, Education… - I am not a cook unless I have to be, so the recipe chapter hit home for me in a different way. However, I can see that chapter being very interesting for my group. I have also dumped in more than I should, then struggled to figure out what to do next. Unfortunately, I’ve just started over again instead of using my somewhat rusty math skills to figure out what to do!
ooops! The first "response" should have been to FMoore instead of NanetteG!
I love the “store” idea melscales suggested. Kids could set up a personal bank account and use the funds to purchase “special time” or whatever each individual situation demands. Currencies could change and students would have to reconfigure their assets. There are a myriad of possibilities.
I chose Chapter 5, balance it, page 46. Most of the problems were multi-step: doing some type of arithmetic, then having to change the units from pounds to ounces, or tons to pounds. The author did a good job of giving helpful hints such as how many ounces in a pound, or how many pounds in a ton. I would consider most of the questions a trifle tricky, mainly in changing units. The questions were definitely not "linear" or just involving a few arithmetic operations. You had to stop and think them through, plan your steps and how to get the correct answer in the correct units. Kind of like putting a little puzzle together. The question that took the most time for me was number two: balancing 4 and 3/8 pounds with 6 and 1/4 pounds. Looks like simple subtraction just turning eighth pounds into quarter pounds, but the author threw in a little monkey wrench because you cannot readily subtract 2/8 from 3/8 at this level. I had to transport myself back to Mrs. Langendorf's 5th grade class and convert both fractions, subtract them, and then convert them back. I was surprised that it came right back to me like an instinct.I had good teachers and consider myself lucky.
Oops!! I meant subtract 3/8 from 2/8!!
kharrellI picked division as I have never been able to easily work division problems or totally understand the thought pattern. I worked the problems on page 140-141. After looking at several of the chapters before the division chapter...I decided to just jump to the einstein level - BAD mistake. It is important to work through the book in order and to go through the different levels. I did not come up with the correct answer the first try. So then I went back to the first of the chapter and worked through the problems. I still came up with the wrong answer...some times I invert numbers - and then finally came up with the answer...after much frustation. (I still think I hate division!) I know I have some students who will skip to the back of the chapter...just wanted to see what would happen if you didn't go through the chapters. For those of us mathematically challenged...the instruction in the early parts of the chapters are important....perhaps my students will be successful...I'll have to try it both ways.In response to gmoore (life) I didn't try the recipes, because I'm always making changes to recipes and doubling or cutting in half, for some reason that chapter was easy...but I don't think of that as math...I agree the kids would have fun with the cooking and eating.In response to Nanette G/Melscales - I wish I had known about the book before, this year our last project (just because a student suggested) was planning a trip around the world. I AVOIDED talking about money exchange because I didn't want to try...but with this chapter...we could have had some fun. I also agree that having some kind of manipulative would make things easier.
I chose to solve the Einstein level question #1 in Chapter 8 on page 80 (“5 cats each had 5 kittens. The Cats and kittens each had 5 fleas. Each flea had 5 tiny spiders on it. How many legs are there all together?”). It was challenging for me because there were multiple calculations involved. I am a visual learner, so I had to sketch out some pictures to keep all the parts straight. First I had to figure out how many of each animal there were, then calculate how many legs each set would have and finally add all the numbers together to get my final answer. The problems in this book are definitely ones I have to take my time to think through – I tended to get the wrong answers when I tried to do quick calculations in my head.
In response to FMoore, I agree that these problems lend themselves to multiple solution techniques which is a great feature for our gifted students. With their different learning styles and creative problem solving strategies, this gives them the freedom to take it and run with it without feeling boxed in to any one particular strategy.
Chapter 8How Manny Legs are There?Page 81 #5How QUICKLY do you turn to multiplication. I started out like we ask the 1st and 2nd graders to do on the Math model. I wrote 1 to 12 in a row and labeled farmers on top. Then I drew a line and wrote 12 next to the farmer and labled Chicks.When it got to eggs, the multiplication clicked in. Then 12 chicks, with 12 eggs each equaled 144 eggs. 144 x 12= 1728 eggs for the the farmers, so then just multiply 1728 x 12 cents = 20736 pennies.Unfortunately I do not see a way to use manipulatives with this Enstein moment/problem. Staring with a basic table diagram was my option. Interesting how you are working with chicks and eggs and then MONEY/Pennies is thrown in there. As I looked back at my worksheet to start to try to write my thought process, I got to the last 12 and could not remember where that 12 came from. Like we tell the children, REREAD the problem and LABEL.
I solved the first problem for Chapter 3: How Much Does It cost? on page 25. To find the cost of the candy bar, I used an algebraic equation: H (hotdog) + C (candy bar) = 5. Then I substituted the info I knew: H is (C+4), so the equation now reads (C+4) + C = 5. From there it was easy to see that the candy bar cost 50 cents (4.50 + .50 = 5.00). Of course, I checked my answer in the back before posting, and liked how the author explains common mistakes made on these questions. In fact, I almost made the mistake he described for this problem!
In response to MillieH,I know what you mean about getting the wrong answer when doing quick calculations in your head! What gets me is that many of our PGP kiddos get the RIGHT answer when they do a quick calculation in their heads! I think getting them to explain how they did it will be a good thing to do.
In response to kharrell,I'm not sure I agree that you have to work through the different levels. I can see a 2nd grade PGP student going straight to Level 3 problems, but a kinder or 1st grader starting on Level 1, depending on their strengths.However, having said that, I did notice on the back of the book that it is intended for 1st - 4th graders, so you may be right!
Chapter 3 - How Much Does it Cost? - Einstein Level p. 25-6I have always been fascinated with money, so I gravitated toward this chapter immediately. I found, as a 1st grade teacher prior to being a Librarian, that my students loved working with money, as well. I really enjoyed working through all the levels, including the Einstein level, of this problem. I definitely had to put pen to paper to solve this problem, and I can see how it would be quite a challenge for my youngest G/T learners. I loved that many of these problems dealt with food, as well. I'd like to present them with the actual items to solve the problems with... you know... hot dogs, candy bars, milk, etc. Tons of concepts here... not just money... a real challenge!!
I just took my granddaughter shopping. She was fascinated when she got change back. (Grammy's purse had a hole in it, she guessed.) So I did Chapter 4, question 1. It reads, "Soccer balls are $9.99 each. If Ruth bought 10 soccer balls and gave the clerk a $100 bill, how much change will she get? I figured that if it was only one cent more to get 10.00 for the balls. So if I multiplied 10.00 by 10 balls I would get 100. If I took the 10 pennies I gained from each ball I bought I would get 10 cents back from 100 dollars.I like what Head Squirrel said about making this actual real life problems for the students to solve.
After spending much time in the car this summer, I chose:Chapter 16: When are We Going to Get There?Einstein LevelAll ProblemsPages 164-165I worked all levels of the problems because I soon found out that I needed to review this particular division concept. On the Einstein level, I missed numbers 2 and 5 which used the exact same concept. Unfortunately, I did those two in my head. When I worked it on paper, I immediately saw my mistake. Again, A+ to the author for including answers to all the problems, not just the evens or odds.I think what I like the most is that while the problems teach a mathematical concept the students are learning a real world skill. I grew up in a very mathematically-minded household. I never remember my father telling us to do certain mathematical problems. Rather he always gave us situations and asked "how would you solve this." Emphasis was on the concept rather than the answer.In response to kharrell and Eleanor, I think this depends. The first time a concept is presented early on in the book it might be important to go in order, but once a student has that initial exposure they might not need to work through all the problems at each level.MillieH, you brought up a good point. While solving a problem in one's head may or may not produce the right answer. The most important point is usually the process. If students show their work through sketches, I believe they are more likely to grasp the concept. To me, this is more important than getting the right answer every time.